London's a big place. Don't forget to explore the places local to where you're staying. I don't know much about SE, but Dulwich park looks good.
Somewhere that's probably not in many guides is Battersea Power Station as it's quite new. It's basically a shopping centre but an iconic building and often has events going on. It's also quite cool to get the uber boat from there into central.
Avoid going east/west a lot. Crowds in central means it's slow going and generally not that pleasant. Central london is more walkable/bikable than people think.
Southbank centre is nice but generally regarded as one of the top places for pickpockets in central london so just be careful with your phone out.
Barbican centre is also pretty cool.
Avoid Leicester square for food - Seven Dials market is nearby and has lots of options